Random Matrix Theory for theWilson-Dirac operator

Mario Kieburg

Department of Physics and AstronomySUNY Stony Brook (NY, USA)

Bielefeld, December 14th, 2011

Outline

I Introduction in Lattice QCD and in the Random MatrixModel

I The Joint Probability DensitiesI The Eigenvalue DensitiesI Conclusions and Outlook

Introduction

in Lattice QCD

and

in the Random Matrix Model

Action of continuums QCD

The partition function of Nf fermionic flavors

Z =

∫exp

−ıSYM(A)− ı

Nf∑j=1

∫ψj(ıD(A)− mj)ψjd4x

D[A, ψ]

The Yang-Mills action of SU(3)

SYM(A) =1

4g2

∫tr FµνFµνd4x

with the field strength tensor

Fµν = ∂µAν − ∂νAµ + g [Aµ,Aν ]

The four components of the vector potential Aµ ∈ su(3) are3 × 3 matrix valued functions.

The continuum Dirac-operator

Fermionic fields ψj are Grassmann variables

⇒ Z =

∫ Nf∏j=1

det(ıD(A)− mj

)exp [−ıSYM(A)]D[A]

The Dirac operator

D(A) = γµ(1ı∂µ + gAµ)

Index-theorem:number of zero modes (index)=topological charge

ν =1

32π2

∫εµνλκtr FµνFλκd4x

Lattice QCD

I space-time becomes discrete with lattice spacing aI vector field Aµ ∈ su(3) replaced by Uµ ∈ SU(3)I Minkowski space → Euclidean space (Wick-rotation t → ıt)

The Dirac matrices in the chiral basis

γ0 =

(0 112

112 0

)→ γ0 =

(0 112

112 0

)γk =

(0 σk

−σk 0

)→ γk =

(0 ıσk

−ıσk 0

)γ5 =

(−112 0

0 112

)→ γ5 =

(112 00 −112

)

Fundamental problem on the lattice

Energy in continuum:

E2 = kµkµ + M20

Energy on lattice:

E2 =sin2(kµa)

a2 + M20 .

Doubler Problem:

kµ →

{kµ

π

a− kµ

one momentum=(24 = 16) particles

The Wilson-Dirac-operator

Main idea:I Make 15 particles in the continuum limit (a → 0) infinitely

heavy.⇒ too inertial, decouple from the systemI Wilson-Dirac operator

DW = D(A) + a∆

∝ γµsin(kµa) +sin2(kµa/2)

a

I Laplace operator ∆I additional effective massI explicitly breaks chiral symmetryI Dirac operator not anymore anti-Hermitian, butγ5-Hermitian

D†W = γ5DWγ5

The ϵ-regime of QCD

I infrared limit of QCDI lattice volume (space-time volume) V → ∞I large Compton wavelength of Goldstone bosons

≫ box size V 1/4

Saddlepoint approximationI spontaneous breaking of chiral symmetryI global Goldstone bosons = Mesons

e.g. Nf = 2, thenSU(2)-integral = zero momentum modes of the three pions

Partition function in ϵ-regime for Nf flavors

Z ∝∫

SU(Nf)exp[L(U)]dµ(U)

∝∑ν∈Z

∫U (Nf)

exp[L(U)]detνUdµ(U)

Lagrangian of the Goldstone bosons:

L(U) =ΣV2

tr (MRU + U†ML)

−VW6a2[tr (U + U†)]2 − VW7a2[tr (U − U†)]2

−VW8a2tr (U2 + U†2)

I index of the Dirac operator: νI masses for right- and left-handed particles: MR/L = M ± Λ

I lattice spacing: aI low energy constants: Σ, W6, W7, W8

I spacetime volume: V

a2-terms of the potential:

VW6a2[tr (U + U†)]2 + VW7a2[tr (U − U†)]2 + VW8a2tr (U2 + U†2)

For SU(2):I Sharpe and Singleton (1998)I Bär, Necco and Schaefer (2009)

For general number of flavors:I Bär, Rupak and Shoresh (2004)I Sharpe (2006)

Simplification

exp[VW6a2[tr (U + U†)]2 + VW7a2[tr (U − U†)]2

]=

12π

∫d [m6, λ7]exp

[−m6

2 + λ72

2

]× exp

[√VW6am6tr (U + U†) +

√VW7aλ7tr (U − U†)

]

I m6, λ7 can be considered as additional masses⇒ omitting the squared trace termsI can be introduced later on

Random Matrix Ensemble

DW =

(aA W

−W † aB

)distributed by

P(DW) ∝ exp[−n

2(tr A2 + tr B2)− n tr WW †

]

I Hermitian matrices A (n × n) and B ((n + ν)× (n + ν)) arethe Wilson-terms ⇒ breaking of chiral symmetry

I complex W (n × (n + ν)) matrixI at a = 0: chGUE describing continuum QCD (Shuryak,

Verbaarschot; 1993)I corresponds to W8 > 0

Damgaard, Splittorff, Verbaarschot (2010)

Microscopic LimitPartition function for Nf flavors

Z ∝∫

d [DW]P(DW)

Nf∏j=1

det(DW + mj112n+ν − λjγ5)

I λ: eigenvalues for D5 = DWγ5 with γ5 = diag (11n,−11n+ν)

I spacetime volume V / matrix dimension n → ∞I fixed parameters:

I ΣV diag (MR, ML) = 2n diag (m + λ,m − λ) = diag (MR, ML)

I√

VW8a =√

n/2a = a

Outcome

Z ∝∫

U (Nf)exp[L(U)]detνUdµ(U)

Lagrangian of the Goldstone bosons:

L(U) = tr (MRU + U†ML)− a2tr (U2 + U†2)

Damgaard, Splittorff, Verbaarschot (2010)

The Joint Probability Densities

Properties of DW and D5

I DW is γ5-Hermitian ⇔ D5 = DWγ5 is HermitianI form invariance:

D5 : V ∈ U (2n + ν), V−1 = V †, compactDW : U ∈ U (n,n + ν), U−1 = γ5U†γ5, non-compact

I diagonalization:

D5 = VxV−1

DW = U

x1 0 0 00 x2 y2 00 −y2 x2 00 0 0 x3

U−1

x , xj , y2 are real diagonal

Diagonalization of DW and D5

I

D5 = VxV−1

x : 2n + ν dim ⇒ pure real spectrum

I Let 0 ≤ l ≤ n

DW = U

x1 0 0 00 x2 y2 00 −y2 x2 00 0 0 x3

U−1

x1: l dim ⇒ real spectrum

x2, y2: n − l dim ⇒ complex conj. eigenvalue pairs x2 ± ıy2

x3: ν + l dim ⇒ real spectrum

Spectral flow of D5 (Example for n = 1 and ν = 2)

Definition of the Joint Probability Density

Let f be arbitrary integrable function invariant underGl(2n + ν,C):∫

f (D5)P(DW)d [DW] =

∫f (x)p5(x)d [x ]

∫f (DW)P(DW)d [DW] =

n∑l=0

∫f (z(l))p(l)

W (z(l))d [z(l)]

=

∫f (z)pW(z)d [z]

where

pW(z) =n∑

l=0

∫p(l)

W (z(l))δ(

z − z(l))

d [z(l)],

z(l) = diag (x1, x2 + ıy2, x2 − ıy2, x3)

Results

For D5 a 2(n + ν) dim Pfaffian:

p5(x) ∝ ∆2n+ν(x)Pf[

g2(xa, xb) xb−1a g1(xa)

−xa−1b g1(xb) 0

]degenerated quark mass mAkemann, Nagao (2011)

For DW a n + ν dim determinant:

pW(z) ∝ ∆2n+ν(z)

× det[

gc(zaR)δ(2)(zaR − z∗

bL) + gr(xaR, xbL)δ(yaR)δ(ybL)

xa−1bL g1(xbL)δ(ybL)

]degenerated source term λγ5Kieburg, Verbaarschot, Zafeiropoulos (2011)

Remark: Vandermonde determinant ∆2n+ν(x) =∏i<j

(xi − xj)

Side remark: Orthogonal Polynomial TheoryA novel mixing of orthogonal and skew-orthogonal polynomials

I orthogonal polynomials from order 0 to ν − 1

⟨pi |pj⟩ ∝ δij

I skew-orthogonal polynomials from order ν

(qν+2i |qν+2j+1) ∝ δij

(qν+2i |qν+2j) = (qν+2i+1|qν+2j+1) = 0

I additional relations

(pi |pj) = (pi |qν+j) = ⟨pi |qν+j⟩ = 0

with

⟨f1|f2⟩ =

∫f1(x)f2(x)g1(x)dx

(f1|f2) =

∫(f1(x1)f2(x2)− f1(x2)f2(x1))G2(x1, x2)d [x ]

I essentially the same system of equations for DW

Side remark: Orthogonal Polynomial Theory

Wish list:I relation to Hermite polynomialsI recursion relationI Christoffel Darboux-like formulaI representation as a matrix integralI Rodrigues formulaI asymptotics in the microscopic limit

The Eigenvalue Densities

Definition

For D5

ρ5(x1) =

∫p5(x)d [x=1]

⇒ one level density ρ5

For DW

ρR(x1R)δ(y1R) +12ρc(z1R) =

∫pW(z)d [z =1R]

ρL(x1L)δ(y1L) +12ρc(z1L) =

∫pW(z)d [z =1L]

⇒ three level densities ρc and

ρr = 2ρR = ρR + ρL − ρχ

ρχ = ρR − ρL

In the Microscopic Limit

For a = 0:

ρ(y) = νδ(y) +y2(J2

ν (y)− Jν−1(y)Jν−2(y))

= νδ(y) + ρ(ν)i (y)

For a = 0:two-fold integrals for ρ5, ρc, ρr, ρχ

Akemann, Damgaard, Kieburg, Nagao, Splittorff, Verbaarschot,Zafeiropoulos (2010/11)

The Density ρc

ρc(z) =

exp[−x2/8a2]√

8πa|y ||z|ρ(ν)i (|z|), a ≪ 1

Θ(8a2 − |x |)16πa2 erf

[|y |√8a

], a ≫ 1

Kieburg, Verbaarschot, Zafeiropoulos (2011)

The Density ρc

ρc(z) =

exp[−x2/8a2]√

8πa|y ||z|ρ(ν)i (|z|), a ≪ 1

Θ(8a2 − |x |)16πa2 erf

[|y |√8a

], a ≫ 1

along y = 5a axis along imaginary axis

Kieburg, Verbaarschot, Zafeiropoulos (2011)

The Density ρχ

ρχ(x) =

ρ(ν)GUE

( x4a

), a ≪ 1

νΘ(8a2 − |x |)π√

64a4 − x2, a ≫ 1

Akemann, Damgaard, Splittorff, Verbaarschot (2010/11)

Number of additional real modes Nadd

Nadd =

∫ρr(x)dx

=

[2(ν + 1)]![(ν + 1)!]2

(a2

2

)ν+1

, a ≪ 1(2π

)3/2

a, a ≫ 1

Kieburg, Verbaarschot,Zafeiropoulos (2011)

Deuzeman, Wenger,Wuilloud (2011)

ρ5 at small Lattice Spacing

ρ5(x) = ρ(ν)GUE

(x − m

4a

)+

|x |√x2 − m2

ρ(ν)i (

√x2 − m2)

= ρχ(x − m) +√

8πa exp[− m2

8a2

]ρc(Re = ım, Im = x)

m = 3, a = 0.2ν ∈ {0,1,2,3,4}

Akemann, Damgaard, Splittorff, Verbaarschot (2010/11)

Lattice Spacing Dependence of ρ5

ρ5(x) =

ρ(ν)GUE

(x − m

4a

)+

|x |√x2 − m2

ρ(ν)i (

√x2 − m2), a ≪ 1

1π

[1 +

cos2 x8a2

], a ≫ 1

m = 3, ν = 1a ∈

{18 ,

14 ,

12 ,1,2

}

Mean Field Limit of ρ5

Scaling:I m → s2mI x → s2xI a → sa

m = 3, ν = 1, a = 0.25s ∈ {1,2,3,4}

Mean Field Limit of ρ5

Scaling:I m → s2mI x → s2xI a → sa

m = 3, ν = 1, a = 0.5s ∈ {1,2,3,4}

Mean Field Limit of ρ5

Scaling:I m → s2mI x → s2xI a → sa

m = 3, ν = 1, a = 1s ∈ {1,2,3,4}

ν Dependence of ρ5 at large Lattice Spacing

Scaling:I m → s2mI x → s2xI a → sa

m = 3, a ∈ {0.5,1}, s = 4

gap in |x | ≤(m2/3 − 4a4/3)3/2

⇒ gap only if m > 8a2

not Aoki Phase Aoki Phase

Damgaard, Splittorff, Verbaarschot (2010)

Conclusions and Outlook

SummaryI joint probability densities of DW and D5

I small a ≪ 1:I broadening of ν formerly zero modes by GUEI Gaussian broadening of ρi(y) for DWI additional real modes are strongly suppressed with

increasing ν

I large a ≫ 1:I finite support of size ≈ a2 along and parrallel to the real

axis for DWI ν independence of DW and D5I mean field limit: gap for ρ5 if m > 8a2

OutlookI non-degenerated massesI higher correlation functionsI comparison with lattice data

Damgaard, Heller, Splittorff (2011)I single eigenvalue distributions

Thank you for your attention!

Collaborators:Gernot AkemannPoul H. DamgaardKim SplittorffJacobus J. M. VerbaarschotSavvas Zafeiropoulos

Appendix

The Density ρr

ρr(x) =

a2ν+1

∫ √2

1pν

(x2

a2 , λ2 − 1

)exp

[− x2

16a2λ2]

dλ, a ≪ 1

Θ(8a2 − |x |)(2π)3/22a

, a ≫ 1

Kieburg, Verbaarschot, Zafeiropoulos (2011)